The Twin Paradox is explored mathematically using the Lorentz coordinate transformations and spacetime diagrams for four different reference frames. Calculated results demonstrate that all observers will agree on the spacetime intervals for the traveling twin and the Earthbound twin, and thus all observers will agree that the traveling twin ages less than the Earthbound twin. The symmetry of time dilation is demonstrated to still be valid, as it must, but in a manner such that no paradox actually arises.